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prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."

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- Thread starter mariouma
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prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."

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cristo

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A lower bound of a non-empty subset A of R is an element d in R with d <= a for all a A.

An element m in R is a greatest lower bound or infimum of A if

m is a lower bound of A and if d is an upper bound of A then m >= d.

An element m in R is a greatest lower bound or infimum of A if

m is a lower bound of A and if d is an upper bound of A then m >= d.

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cristo

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-A:={-x:x∈A}, and use the completeness axiom to find the least upper bound on -A. The relationship between A and -A should help you find the greatest lower bound of A.

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this is not number theory, but it is instead mathematical analysis.

it is a very basic result that has been used to construct the real numbers and i think you will find it in any standard intro to analysis textbook (i personally recommend principles of mathematical analysis by Walter Rudin).

hope it helps

Aditya Babel

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HallsofIvy

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adityab88 is, I think, assuming that you have to prove that completenss axiom from the definition of the real numbers. It very easy to do that from the Dedekind cut definition of the real numbers.

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adityab88 is, I think, assuming that you have to prove that completenss axiom from the definition of the real numbers. It very easy to do that from the Dedekind cut definition of the real numbers.

precisely. dedekind cuts or even cauchy sequences can be used prove such a statement.

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